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Theorem cnvcnvss 5313
Description: The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
cnvcnvss  |-  `' `' A  C_  A

Proof of Theorem cnvcnvss
StepHypRef Expression
1 cnvcnv 5311 . 2  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
2 inss1 3591 . 2  |-  ( A  i^i  ( _V  X.  _V ) )  C_  A
31, 2eqsstri 3407 1  |-  `' `' A  C_  A
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 2993    i^i cin 3348    C_ wss 3349    X. cxp 4859   `'ccnv 4860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-xp 4867  df-rel 4868  df-cnv 4869
This theorem is referenced by:  funcnvcnv  5497  foimacnv  5679  cnvfi  7616  structcnvcnv  14206  strlemor1  14286  mvdco  15972  fcnvgreu  26013  cnvct  26037
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